Journal
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
Volume 42, Issue 6, Pages 2610-2625Publisher
SIAM PUBLICATIONS
DOI: 10.1137/10078503X
Keywords
global well-posedness; viscoelastic fluid system; bounded domains
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Funding
- Morningside Center of Mathematics of the Chinese Academy of Sciences (CAS)
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In this paper, the initial boundary value problem of the incompressible viscoelatic fluid system is considered. In the contrast to the previous work [Y. Chen and P. Zhang, Comm. Partial Differential Equations, 31 (2006), pp. 1793-1810], [F. H. Lin and P. Zhang, Comm. Pure Appl. Math., 61 (2008), pp. 539-558], [F. H. Lin, C. Liu, and P. Zhang, Comm. Pure Appl. Math., 18 (2005), pp. 1437-1471], and [Z. Lei, C. Liu, and Y. Zhou, Arch. Ration. Mech. Anal., 188 (2008), pp. 371-398], our main ingredient to solve the problem is to reformulate the original system along the particle trajectory. It is shown that in the so-called Lagrangian coordinate, the system is decoupled, and the well-posedness problem is reduced to the solvability of standard damped-wave equations with a no-slip boundary condition. Finally, we establish the global existence and exponential decay estimate for the equivalent system with the small and smooth initial data.
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